Set Theory
2.0 Representation of set
2.0 Representation of set
There are two methods of representing a set:
i. Roster method representation
In this method each element in the set is written.
The set is represented with elements within braces.The elements can be listed in any order. The elements are separated by comas.
The element must not be repeated.
i.e. $$\begin{equation} \begin{aligned} A = \{ 11,7,9,5\} \\ B = \{ Vineeth,\,Sudha\,,\;Karna\} \\\end{aligned} \end{equation} $$
are some examples of a set in roster form.
Illustration 3. Represent the following sets in the roster form
a. Collection of vowels in the alphabet of English language
b. The integer factors of $9$
c. Natural numbers less than $10$
d. The set of squares of integers
e. Set of all alphabets used to form "PROPORTIONALITY"
Solution:
| SET | ELEMENTS IN ROSTER FORM |
| a. Collection of vowels in the alphabet of English language | The elements are $a, e, i, o, u$ The elements in roster form is $\{ a,\;e,\;i,\;o,\;u\} $ |
| b. The positive integer factors of $9$ | The elements are $1,\,3,9$ The elements in roster form is $\{ \,1,\,3,9\,\} $ |
| c. Natural numbers less than $10$ | The elements are $1, 2, 3, 4, 5, 6, 7, 8, 9$ The elements in roster form is $\{ 1,2,3,4,5,6,7,8,9\} $ |
| d. The set of squares of integers | Thus the elements are $1,\;4,\;9,\;16,\;25,.........$ The elements in roster form is $\{ 1,\;4,\;9,\;16,\;25,.........\} $ |
| e. Set of all alphabets used to form "PROPORTIONALITY" | The elements are $P,\,R,\,O,\;T,\;I,\;N,\;A,\;L,\;Y$ The elements in roster form is $\{ A,\;I,\;L,\;N,\,O,\;P,\,R,T,Y\} $ |
ii. Set-builder form
In this method, the set is represented by stating the properties within braces, which are satisfied by all the elements of the set.
The set is represented in words within braces.
In this form, $:$ is read as 'such that'.
A common property representing every element of the set is written after the '$:$'.
i.e. $$\begin{equation} \begin{aligned} A = \{ x:x = 2n - 1,\,n \ge 1\;,\;n \in N\} \\ B = \{ x:1 < x < 18,x\;is\;a\;multiple\;of\;3\} \\\end{aligned} \end{equation} $$
are some examples of a set in set-builder form.
Illustration 4. Represent the following sets in the set-builder form
a. The set of reciprocals of natural numbers
b. Set of all odd natural numbers
c. The set of alphabets used to form "ANTARCTICA"
d. The collection of numbers greater than $-10$ and less than $5$
e. The set of square numbers
Solution:
| SET | ELEMENTS IN SET-BUILDER FORM |
| a. The set of reciprocals of natural numbers | $\{ x\;:\;x = {1 \over n},\;n \in N\} $ or $\{ {1 \over x}:\;x \in N\} $ |
| b. Set of all odd natural numbers | $\{ x\;:\;x = 2n - 1,\;n \in N\} $ |
| c. The set of alphabets used to form "ANTARCTICA" | $\{ x\;:\;x\;\;is\;a\;letter\;in\;the\;word\;ANTARCTICA\} $ |
| d. The collection of numbers greater than $-10$ and less than $5$ | $\{ x\;:\; - 10 < x < 5,\;x \in Z\} $ |
| e. The set of square numbers | $\{ x\;:\;x = {n^2},\;n \in Z\} $ or $\{ {x^2}\;:\;x \in Z\} $ |
Illustration 5. Represent the following sets in the roster form and set builder form
a. The set of all natural numbers between $11$ and $15$
b. The set of primary colors
c. The set of all states of India beginning with the letter $A$
d. The set of roots of natural numbers
e. The set of letters forming the word "FRICTION"
Solution:
| SET | ELEMENTS IN ROSTER FORM | ELEMENTS IN SET-BUILDER FORM |
| a. The set of all natural numbers between $11$ and $15$ | $\{ 12,13,14\} $ | $\{ x:\;11 < x < 15,\,\,x \in N\} $ |
| b. The set of primary colors | $\{ red,\;yellow,\;blue\} $ | $\{ x:\;x\;is\,a\;primary\,color\} $ |
| c. The set of all states of India beginning with the letter $M$ | $\{ Maharashtra, \;Madhyapradesh, \;Mizoram, \;Manipur,\;Meghalaya\} $ | $\{ x:\;x\;is\,an\;Indian\;state\;beginning\;with\;M\} $ |
| d. The set of roots of natural numbers | $\{ 1,\sqrt 2 ,\sqrt 3 ,\;2,\sqrt 5 \;,\sqrt 6 ,......\} $ | $\{ x:\;x = \sqrt n ,\,n \in N\} $ |
| e. The set of letters forming the word "FRIENDS" | $\{ F,R,I,E,N,D,S\} $ | $\{ x:\;x\;is\,a\;letter\;in\;the\;word\;FRIENDS\} $ |
